An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feasible augmented La-grangian,which can be solved by the augmented Lagrangian method in an infinite dimensional setting.Based on this,by expressing the primal and dual variables with two individual deep neural network functions,we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimiza-tion method together with a projection technique.Compared to the traditional penalty method,the new method admits two main advantages:i)the choice of the penalty parameter isflexible and robust,and ii)the numerical solution is more accurate in the same magnitude of computational cost.As typical applications,we apply the new ap-proach to solve elliptic problems and(nonlinear)eigenvalue problems with essential boundary conditions,and numerical experiments are presented to show the effective-ness of the new method.

Binary Level Set Methods for Dynamic Reservoir Characterization by Operator Splitting Scheme

In this paper,operator splitting scheme for dynamic reservoir characterization by binary level set method is employed.For this problem,the absolute permeability of the two-phase porous medium flow can be simulated by the constrained augmented Lagrangian optimization method with well data and seismic time-lapse data.By transforming the constrained optimization problem in an unconstrained one,the saddle point problem can be solved by Uzawas algorithms with operator splitting scheme,which is based on the essence of binary level set method.Both the simple and complicated numerical examples demonstrate that the given algorithms are stable and efficient and the absolute permeability can be satisfactorily recovered.